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arXiv:cond-mat/0511727 · Physical Review E

Application of the Hilbert space average method on heat conduction models

Mathias Michel A, Jochen Gemmer B, Günter Mahler A

Published Jan 3, 2006• 17 citations
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Abstract

We analyze closed one-dimensional chains of weakly coupled many level systems, by means of the so-called Hilbert space average method (HAM). Subject to some concrete conditions on the Hamiltonian of the system, our theory predicts energy diffusion with respect to a coarse-grained description for almost all initial states. Close to the respective equilibrium, we investigate this behavior in terms of heat transport and derive the heat conduction coefficient. Thus, we are able to show that both heat (energy) diffusive behavior as well as Fourier's law follows from and is compatible with a reversible Schrödinger dynamics on the complete level of description.

DOI
10.1103/physreve.73.016101

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We analyze closed one-dimensional chains of weakly coupled many level systems, by means of the so-called Hilbert space average method (HAM). Subject to some concrete conditions on the Hamiltonian of the system, our theory predicts energy diffusion with respect to a coarse-grained description for almost all initial states. Close to the respective equilibrium, we investigate this behavior in terms of heat transport and derive the heat conduction coefficient. Thus, we are able to show that both heat (energy) diffusive behavior as well as Fourier's law follows from and is compatible with a reversible Schrödinger dynamics on the complete level of description.
In this work we analyze the time evolution of randomly coupled systems. We present a derivation for the coarse grained dynamics (dynamics of Macro-Observables) starting directly from the Schrödinger equation. In doing this derivation we need to take some special limits such as large size of the Hilbert space, large time and weak coupling. We find that the dynamics in these limits converge to those given by rate equations, with Fermi's Golden rule fulfilled, or the diffusion equation. We also analyze how typical the dynamics are, that is how far can we trust the derivation for a specific chosen coupled system of the random ensemble.
The origin of classical predictability is investigated for the one dimensional harmonic chain considered as a closed quantum mechanical system. By comparing the properties of a family of coarse-grained descriptions of the chain, we conclude that local coarse grainings in this family are more useful for prediction than nonlocal ones. A quantum mechanical system exhibits classical behavior when the probability is high for histories having the correlations in time implied by classical deterministic laws. But approximate classical determinism holds only for certain coarse grainings and then only if the initial state of the system is suitably restricted. Coarse grainings by the values of the hydrodynamic variables (integrals over suitable volumes of densities of approximately conserved quantities) define the histories usually used in classical physics. But what distinguishes this coarse graining from others? This paper approaches this question by analyzing a family of coarse grainings for the linear harmonic chain. At one extreme in the family the chain is divided into local groups of N atoms. At the other extreme the N atoms are distributed nonlocally over the whole chain. Each coarse graining follows the average (center of mass) positions of the groups and ignores the ``internal'' coordinates within each group, these constituting a different environment for each coarse graining. For an initial condition where long wavelength modes are excited and short wavelength modes are distributed thermally we find that the coarse-grained positions obey deterministic equations of motion accompanied by noise. The noise is greater the more nonlocal the coarse graining. Further, the deterministic equations require more time steps to evolve over a given time interval for the nonlocal coarse grainings than for the local ones. A continuum limit is possible only for the near local coarse grainings. For parameters of the model characteristic of realistic situations these features strongly favor the local coarse grainings over the nonlocal ones for prediction. Each of these differences can be traced to the approximate conservation of the local center of mass momentum. We then consider the chain quantum mechanically and show that, for realistic parameters, all the coarse grainings decohere rapidly compared to dynamical time scales. We conclude that noise, decoherence, and computational complexity favor locality over nonlocality for deterministic predictability.
We investigate energy transport in several two-level atom or spin-1∕21∕21∕2 models by a direct coupling to heat baths of different temperatures. The analysis is carried out on the basis of a recently derived quantum master equation which describes the nonequilibrium properties of internally weakly coupled systems appropriately. For the computation of the stationary state of the dynamical equations, we employ a Monte Carlo wave-function approach. The analysis directly indicates normal diffusive or ballistic transport in finite models and hints toward an extrapolation of the transport behavior of infinite models.
The heat conductivity of nonperiodic quantum Ising chains whose ends are connected with heat baths at different temperatures are studied numerically by solving the Lindblad master equation. The chains are subjected to a uniform transverse field h, while the exchange coupling J{m} between the nearest-neighbor spins takes the two values J{A} and J{B} arranged in Fibonacci, generalized Fibonacci, Thue-Morse, and period-doubling sequences. We calculate the energy-density profile and energy current of the resulting nonequilibrium steady states to study the heat-conducting behavior of finite but large systems. Although these nonperiodic quantum Ising chains are integrable, it is clearly found that energy gradients exist in all chains and the energy currents appear to scale as the system size ~N{α}. By increasing the ratio of couplings, the exponent α can be modulated from α > -1 to α < -1 corresponding to the nontrivial transition from the abnormal heat transport to the heat insulator. The influences of the temperature gradient and the magnetic field to heat conduction have also been discussed.